Quant Boosters  Ravi Handa  Set 1

Wilson's Theorem says For a prime number 'p'
Rem [ (p1)! / p] = p1This can be extended to say,
Rem [ (p2)! / p] = 1Let us use that here. We need to find out Rem [97! / 101] = r
We know from the above theorem,
Rem [99! / 101] = 1
=> Rem [99 * 98 * 97! / 101] = 1
=> Rem [ (2)*(3)*r / 101] = 1
=> Rem [6r / 101] = 1
=> 6r = 101k + 1
We need to think of a value of k, such that 101k + 1 is divisible by 6.
If we put, k = 1, we get 101 + 1 = 102, which is divisible by 6.
=> 6r = 102
=> r = 17

Q21) If a person starts writing all 4 digit numbers, how many times has he written the digit 2?

Let us call the 4 digit number as abcd
When a = 2, b / c / d can be filled in 10 ways each.
So, 2 will appear in place of 'a' 10 * 10 * 10 = 1000 times
When b = 2, a can be filled in 9 ways whereas c / d can be filled in 10 ways each.
So, 2 will appear in place of 'b' 9 * 10 * 10 = 900 times
Similarly, 2 will appear in place of 'c' 900 times and in place of 'd' 900 times.
Total number of times digit '2' will appear = 1000 + 900 + 900 + 900 = 3700 times

Q22) What is the number of digits in 2^2009?

Using Logarithms is probably the fastest and the best way to get to the answer.
If Log(x) = a.bcde, number of digits in x is a + 1
Log (2^2009) = 2009 Log(2) = 2009 * 0.3010 = 604.709
=> 2^2009 has 604 + 1 = 605 digits

Q23) How many digits are there in (2ABC)^4 where 2ABC is a 4 digit number?

We need to find out the number of digits in (2ABC)^4
The smallest such number will be 2000^4 = 16*10^12
This will have 14 digits.The biggest such number will be 2999^4. While that is hard to find out, we can find out 3000^4 easily.
3000^4 = 81*10^12. It has 14 digits
=> 2999^4 will also have 14 digits.As the smallest and the biggest number of the format (2ABC)^4 has 14 digits, we can say that there are 14 digits in (2ABC)^4

Q24) The product of two numbers is 7168 and their highest common factor is 16. How many pairs of numbers are possible such that the above condition is satisfied?

Let us say that the numbers are x and y.
Both of them are multiples of 16 as their HCF is 16
So, we can say that they are 16a and 16b.Also, HCF (a,b) = 1. They have no factor in common otherwise the HCF of 16a and 16b won't be 16. In other words, a and b are coprime to each other.
x * y = 7168
=> 16a * 16b = 7168
=> a * b = 28
Now, we need to split 28 into coprime factors
28 = 1 * 28 or 4 * 7
=> (x, y) = (16, 448) or (64, 112)So, there are two pairs that satisfy the given conditions.
Note: I have considered only positive numbers for the calculation shown above. If you wish to consider negative numbers as well, the answer would double

Q25) 31 consecutive leaves were torn off. which of following could be the sum of 62 page numbers on these leaves?
a) 1955
b) 2201
c) 2079
d) none

From the leaves that are torn off, the first page number should be odd.
Let us assume it is 2a + 1.
Since there are total of 62 pages which are torn off, the last page number will be 2a + 62
Sum of 'n' terms in an Arithmetic Progression = n/2[First term + Last Term]
=> Sum of page numbers = 62/2 [2a + 1 + 2a + 62]
=> Sum of page numbers = 31(4a + 63)Now, let's look at the options
Option (a) 1955 is ruled out because it is not divisible by 31Option (b) 2201
=> 31(4a + 63) = 2201
=> 4a + 63 = 2201/31 = 71
=> 4a = 71  63 = 8
=> a = 2
=> 2201 is a valid valueOption (c) 2079 is ruled out because it is not divisible by 31

Q26) How many times is a key of a typewriter pressed in order to type the first 299 natural numbers by pressing the space bar once between any two successive natural numbers?

Number of 1 digit natural numbers {1, 2, 3 ... 9}= 9
Keystrokes required to type them = 9*1 = 9Number of 2 digit natural numbers {10, 11, 12 ... 99} = 90
Keystrokes required to type them = 90*2 = 180Number of 3 digit natural numbers {100, 101, 102 ... 299} = 200
Keystrokes required to type them = 200*3 = 600Number of spaces between the numbers = 298
Keystrokes required to type them = 298*1 = 298Total keystrokes required = 9 + 180 + 600 + 298 = 1087

Q27) Which number is greater, 80^50 or 60^54?

To solve this, it would help if you know Log(2) = 0.3010 and Log(3) = 0.4771
If we have to compare a and b, we can compare Log(a) and Log(b).Log (60^54) = 54 Log (60) = 54 [Log (2) + Log(3) + Log(10)] = 54[0.3010 + 0.4771 + 1] = 54 * 1.771 = 95.634
Log (80^50) = 50 Log(80) = 50 [Log(8) + Log(10)] = 50[3 Log(2) + 1] = 50[3*0.3010 + 1] = 50 * 1.9030 = 95.15
So, Log (60^54) > Log (80^50) => 60^54 > 80^50

Q28) How many 6 digit numbers can be made using digits 1, 2, 3, 4, 5, and 6 without repetition such that the hundred digit is greater than the ten digit, and the ten is greater than the one digit?

Let us assume that the number that we have is abcdef
The condition given is d > e > fLet us first select three digits for a, b, and c.
From the given 6 digits (1,2,3,4,5,6), 3 digits for a,b, and c can be selected in 6C3 = 6!/3!3! = 20 ways.
These three selected digits can be arranged on three positions of a, b, and c in 3! = 6 waysThe remaining 3 digits are automatically selected for d, e, and f.
So, we have 1 way of selecting them.The biggest digit will be allocated to 'd'.
The second biggest digit will be allocated to 'e'.
The third biggest digit will be allocated to 'f'
So, we have 1 way of arranging them on the three positions of d, e, and f.Total ways = 20 * 6 * 1 * 1 = 120

Q29) A and B started at a time towards each other. After crossing each other, they took 16hrs, 25hrs respectively to reach their destinations. If A had 40kmph speed, what is the speed of B?

Let us say that A and B started from point P and point Q respectively and met at point R in the middle after time 't'.
Speed of A is given to us as 40 kmph.
Let us assume the speed of B to be 's'.
If we consider the distance PR, it was covered by A in 't' hours and B in 25 hours.
=> PR = 40 * t = s * 25If we consider the distance RQ, it was covered by B in 't' hours and A in '16' hours.
=> RQ = s * t = 40 * 16Let us divide the two equations. We will get
40t/st = 25s/640
=> 40/s = 5s/128
=> s^2 = 40 * 128/5 = 8 * 128 = 1024
=> Speed of B = s = 32 kmph

Q30) What degree occurs at a time 3:30 in the clock?